Question

If P is a point on the hyperbola whose axis are equal, prove that SP×SP'=CP²

Answer 1

For an hyperbola if the length of semi transverse and semi conjugate axes are equal.

Then a = b

∴ Equation of the given hyperbola is

x² – y² = a²  ......(1)

Let coordinates of any point P on hyperbola be (α, β). Since P lies on (1)

∴ α² – β² = a²    ......(2)

Now SP² .S'P² = (2a² + a² + β²)² – 8a²α²

= 4a⁴ + 4a² (α² + β²) + (α² + β²)² – 8a²α²

= 4a² (a²  – 2α²) + 4a² (α² + β²) + (α²+ β²)²

= 4a² (α²  –  β² – 2α²) + 4a² (α² + β²) + (α²+ β²)²

= (α²+ β²)² = CP⁴

∴ SP. S'P = CP²